Bird J. Electrical Circuit Theory and Technology
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828 Electrical Circuit Theory and Technology
Thus Z
0
1
D
4mZ
2
1
Z
2
4Z
1
Z
2
C Z
2
1
1 m
2
i.e., Z
1
=
4mZ
1
Z
2
4Z
2
Y Z
1
.1 − m
2
/
42.31
An impedance mZ
1
in parallel with an impedance 4mZ
2
/1 m
2
gives
(using (product/sum)):
mZ
1
4mZ
2
1 m
2
mZ
1
C
4mZ
2
1 m
2
D
mZ
1
4mZ
2
mZ
1
1 m
2
C 4mZ
2
D
4mZ
1
Z
2
4Z
2
C Z
1
1 m
2
Hence the expression for Z
0
1
(equation (42.31)) represents an impedance
mZ
1
in parallel with an impedance 4m/1 m
2
Z
2
(d) Low-pass ‘m-derived’ sections
The ‘m-derived’ low-pass T section is shown in Figure 42.43(a) and is
derived from Figure 42.13(a), Figure 42.41 and equation (42.30). If Z
2
represents a pure capacitor in Figure 42.41(a), then Z
2
D 1/ωC.
A capacitance of value mC shown in Figure 42.43(a) has an impedance
1
ωmC
D
1
m
1
ωC
D
Z
2
m
as in equation (42.30).
The ‘m-derived’ low-pass section is shown in Figure 42.43(b) and is
derived from Figure 42.13(b), Figure 42.42 and from equation (42.31).
Note that a capacitance of value
1 m
2
4m
C has an impedance of
1
ω
1 m
2
4m
C
D
4m
1 m
2
1
ωC
D
4m
1 m
2
Z
2
mL
2
mC
mL
(a)
(b)
mL
2
1−m
2
4 m
L
1−m
2
4 m
C
mC
2
mC
2
Figure 42.43
where Z
2
is a pure capacitor.
In Figure 42.43(a), series resonance will occur in the shunt arm at a
particular frequency—thus short-circuiting the transmission path. In the
prototype, infinite attenuation is obtained only at infinite frequency (see
Figure 42.25).
In the m-derived section of Figure 42.43(a), let the frequency of infinite
attenuation be f
1
, then at resonance: X
L
D X
C
i.e., ω
1
1 m
2
4m
L D
1
ω
1
mC
from which, ω
2
1
D
1
mC
1 m
2
4m
L
D
4
LC1 m
2
Filter networks 829
From equation (42.2),
4
LC
D ω
2
c
, thus ω
2
1
D
ω
2
c
1 m
2
,
where ω
c
D 2f
c
, f
c
being the cut-off frequency of the prototype.
Hence ω
1
D
ω
c
p
1 m
2
42.32
Rearranging gives: ω
2
1
1 m
2
D ω
2
c
ω
2
1
m
2
ω
2
1
D ω
2
c
m
2
D
ω
2
1
ω
2
c
ω
2
1
D 1
ω
2
c
ω
2
1
i.e.,
m =
1 −
f
c
f
∞
2
42.33
In the m-derived section of Figure 42.43(b), resonance occurs in the
parallel arrangement comprising the series arm of the section when
ω
2
D
1
mL
1 m
2
4m
C
, when ω
2
D
4
LC1 m
2
as in the series resonance case (see Chapter 28).
Thus equations (42.32) and (42.33) are also applicable to the low-pass
m-derived section.
In equation (42.33), 0 < m < 1, thus f
∞
> f
c
.
The frequency of infinite attenuation f
1
can be placed anywhere within
the attenuation band by suitable choice of the value of m; the smaller m
is made the nearer is f
1
to the cut-off frequency, f
c
.
Problem 15. A filter section is required to have a nominal
impedance of 600 , a cut-off frequency of 5 kHz and a frequency
of infinite attenuation at 5.50 kHz. Design (a) an appropriate ‘m-
derived’ T section, and (b) an appropriate ‘m-derived’ section.
Nominal impedance R
0
D 600 , cut-off frequency, f
c
D 5000 Hz and
frequency of infinite attenuation, f
1
D 5500 Hz. Since f
1
>f
c
the
filter section is low-pass.
830 Electrical Circuit Theory and Technology
From equation (42.33),
m D
1
f
c
f
1
2
D
1
5000
5500
2
D 0.4166
For a low-pass prototype section:
from equation (42.6), capacitance, C D
1
R
0
f
c
D
1
6005000
0.106
µF
and from equation (42.7), inductance, L D
R
0
f
c
D
600
5000
38.2mH
(a) For an ‘m-derived’ low-pass T section:
From Figure 42.43(a), the series arm inductances are each
mL
2
D
0.416638.2
2
D 7.957 mH,
and the shunt arm contains a capacitor of value mC,
i.e., 0.41660.106 D 0.0442 mF or 44.2 nF, in series with an
inductance of
value
1 m
2
4m
L D
1 0.4166
2
40.4166
38.2,
i.e., 18.95 mH
The appropriate ‘m-derived’ T section is shown in Figure 42.44.
(b) For an ‘m-derived’ low-pass section:
From Figure 42.43(b) the shunt arms each contain capacitances
equal to mC/2,
i.e.,
0.41660.106
2
D 0.0221 mFor22.1nF,
7.957 mH 7.957 mH
44.2 nF
18.95 mH
Figure 42.44
and the series arm contains an inductance of value mL,
i.e., 0.416638.2 D 15.91 mH in parallel with a capacitance of
value
1 m
2
4m
C D
1 0.4166
2
40.4166
0.106
D 0.0526 mFor52.6nF
The appropriate ‘m-derived’ section is shown in Figure 42.45.
15.91 mH
22.1 nF
52.6 nF
22.1 nF
Figure 42.45
Filter networks 831
(e) High-pass ‘m-derived’ sections
Figure 42.46(a) shows a high-pass prototype T section and Figure 42.46(b)
shows the ‘m-derived’ high-pass T section which is derived from
Figure 42.16(a), Figure 42.41 and equation (42.30).
Z
1
2
Z
1
2
Z
2
2C 2C
L
2C
m
2C
m
L
m
4m
1 − m
2
C
(a)
(b)
Figure 42.46
Figure 42.47(a) shows a high-pass prototype section and
Figure 42.47(b) shows the ‘m-derived’ high-pass section which is
derived from Figure 42.16(b), Figure 42.42 and equation (42.31). In
Figure 42.46(b), resonance occurs in the shunt arm when:
ω
1
L
m
D
1
ω
1
4m
1 m
2
C
i.e., when ω
2
1
D
1 m
2
4LC
D ω
2
c
1 m
2
from equation (42.14)
i.e., ω
1
D ω
c
p
1 m
2
42.34
Hence
ω
2
1
ω
2
c
D 1 m
2
Z
1
2Z
2
2Z
2
(a)
C
2L 2L
2L
m
C
m
4 m
1 − m
2
L
2L
m
(b)
Figure 42.47
832 Electrical Circuit Theory and Technology
from which,
m =
1 −
f
∞
f
c
2
42.35
For a high-pass section, f
1
<f
c
.
It may be shown that equations (42.34) and (42.35) also apply to the
‘m-derived’ section shown in Figure 42.47(b).
Problem 16. Design (a) a suitable ‘m-derived’ T section, and
(b) a suitable ‘m-derived’ section having a cut-off frequency of
20 kHz, a nominal impedance of 500 and a frequency of infinite
attenuation 16 kHz.
Nominal impedance R
0
D 500 , cut-off frequency, f
c
D 20 kHz and the
frequency of infinite attenuation, f
1
D 16 kHz. Since f
1
<f
c
the filter
is high-pass.
From equation (42.35), m D
1
f
1
f
c
2
D
1
16
20
2
D 0.60
For a high-pass prototype section:
From equation (42.18), capacitance,
C D
1
4R
0
f
c
D
1
450020000
7.958 nF
and from equation (42.19), inductance,
L D
R
0
4f
c
D
500
420 000
1.989 mH
(a) For an ‘m-derived’ high-pass T section:
From Figure 42.46(b), each series arm contains a capacitance of
value 2C/m, i.e., 2(7.958)/0.60, i.e., 26.53 nF, and the shunt arm
contains an inductance of value L/m, i.e., 1.989/0.60 D 3.315 mH
in series with a capacitance of value
4m
1 m
2
C i.e.,
40.60
1 0.60
2
7.958 D 29.84 nF
A suitable ‘m-derived’ T section is shown in Figure 42.48.
(b) For an ‘m-derived’ high pass section:
From Figure 42.47(b), the shunt arms each contain inductances equal
to 2L/m, i.e., (2(1.989)/0.60), i.e., 6.63 mH and the series arm
26.53 nF 26.53 nF
3.315 mH
29.84 nF
Figure 42.48
Filter networks 833
13.26 nF
7.459 mH
6.63 mH 6.63 mH
Figure 42.49
contains a capacitance of value C/m, i.e., 7.958/0.60 D 13.26 nF
in parallel with an inductance of value 4m/1 m
2
L,
i.e.,
40.60
1 0.60
2
1.989 7.459 mH
A suitable ‘m-derived’ section is shown in Figure 42.49.
Further problems on ‘m-derived’ filter sections may be found in
Section 42.10, problems 19 to 22, page 839
42.9 Practical composite
filters
In practise, filters to meet a given specification often have to comprise a
number of basic networks. For example, a practical arrangement might
consist of (i) a basic prototype, in series with (ii) an ‘m-derived’ section,
with (iii) terminating half-sections at each end. The ‘m-derived’ section
improves the attenuation immediately after cut-off, the prototype improves
the attenuation well after cut-off, whilst the terminating half-sections are
used to obtain a constant match over the pass-band.
Figure 42.50(a) shows an ‘m-derived’ low-pass T section, and
Figure 42.50(b) shows the same section cut into two halves through AB,
each of the two halves being termed a ‘half-section’. The ‘m-derived’ half
section also improves the steepness of attenuation outside the pass-band.
Z
0T
Z
0T
mL
2
mL
2
mC
1 − m
2
4 m
(a)
Z
0T
mC
2
mC
2
L
Z
0T
Z
0T
(b)
1 − m
2
2 m
L
1 − m
2
2 m
L
mL
2
mL
2
A
B
Z
0T
Figure 42.50
As shown in Section 42.8, the ‘m-derived’ filter section is based on a
prototype which presents its own characteristic impedance at its terminals.
Hence, for example, the prototype of a T section leads to an ‘m-derived’
T section and Z
0T
D Z
0Tm
where Z
0T
is the characteristic impedance
of the prototype and Z
0Tm
is the characteristic impedance of the ‘m-
derived’ section. It is shown in Figure 42.24 that Z
0T
has a non-linear
characteristic against frequency; thus Z
0Tm
will also be non-linear.
Since from equation (42.9), Z
0
D Z
1
Z
2
/Z
0T
, then the characteristic
impedance of the ‘m-derived’ section,
Z
0m
D
Z
0
1
Z
0
2
Z
0Tm
D
Z
0
1
Z
0
2
Z
0T
834 Electrical Circuit Theory and Technology
where Z
0
1
and Z
0
2
are the equivalent values of impedance in the ‘m-derived’
section.
From Figure 42.41, Z
0
1
D mZ
1
and from equation (42.30),
Z
0
2
D
Z
2
m
C
1 m
2
4m
Z
1
Thus Z
0m
D
mZ
1
Z
2
m
C
1 m
2
4m
Z
1
Z
0T
D
Z
1
Z
2
Z
0T
1 C
1 m
2
4Z
2
Z
1
42.36
or Z
0p.m/
= Z
0p
1 Y
1 − m
2
4Z
2
Z
1
42.37
Thus the impedance of the ‘m-derived’ section is related to the impedance
of the prototype by a factor of [1 C1 m
2
/4Z
2
Z
1
] and will vary as m
varies.
When m D 1,Z
0m
D Z
0
When m D 0,Z
0m
D
Z
1
Z
2
Z
0T
!
1 C
Z
1
4Z
2
"
from equation (42.36)
D
1
Z
0T
Z
1
Z
2
C
Z
2
1
4
However from equation (42.8), Z
1
Z
2
C
Z
2
1
4
D Z
2
0T
Hence, when m D 0,Z
0m
D
Z
2
0T
Z
0T
D Z
0T
Thus the characteristic of impedance against frequency for m D 1and
m D 0 shown in Figure 42.51 are the same as shown in Figure 42.24.
Further characteristics may be drawn for values of m between 0 and 1 as
shown.
It is seen from Figure 42.51 that when m D 0.6 the impedance is prac-
tically constant at R
0
for most of the pass-band. In a composite filter,
‘m-derived’ half-sections having a value of m D 0.6 are usually used at
each end to provide a good match to a resistive source and load over the
pass-band.
Figure 42.51 shows characteristics of ‘m-derived’ low-pass filter
sections; similar curves may be constructed for m-derived high-pass filters
with the two curves shown in Figure 42.29 representing the limiting
values of m D 0andm D 1.
The value of m needs to be small for the frequency of input attenuation,
f
1
, to be close to the cut-off frequency, f
c
. However, it is not practical
Filter networks 835
Z
0
R
0
Z
OT
Z
0p
= Z
0p(m)
m = 1
m = 0.8
m = 0.6
m = 0.4
Z
OT
= Z
0p
(m)
m = 0
0 f
c
Frequency
Pass band
Attenuation
band
Nominal
impedance
Z
0π(m)
Figure 42.51
to make m very small, below 0.3 being very unusual. When m D 0.3,
f
1
³ 1.05f
c
(from equation (42.32)) and when m D 0.6, f
1
D 1.25f
c
.
The effect of the value of m on the frequency of infinite attenuation is
shown in Figure 42.52 although the ideal curves shown would be modified
a little in practise by resistance losses.
Attenuation
m = 0.3
m = 0.6
m = 1
f
c
1.05f
c
1.25f
c
Frequency
Figure 42.52
Problem 17. It is required to design a composite filter with a
cut-off frequency of 10 kHz, a frequency of infinite attenuation
11.8 kHz and nominal impedance of 600 . Determine the compo-
nent values needed if the filter is to comprise a prototype T section,
an ‘m-derived’ T section and two terminating ‘m-derived’ half-
sections.
836 Electrical Circuit Theory and Technology
R
0
D 600 , f
c
D 10 kHz and f
1
D 11.8 kHz. Since f
c
<f
1
the filter
is a low-pass T section.
For the prototype:
From equation (42.6), capacitance,
C D
1
f
c
R
0
D
1
10 000600
0.0531
µF,
and from equation (42.7), inductance,
L D
R
0
f
c
D
600
10 000
19 mH
Thus, from Figure 42.13(a), the series arm components are
L/2 D 19/2 D 9.5mHand the shunt arm component is 0.0531 mF.
For the ‘m-derived’ section:
From equation (42.33),
m D
1
f
c
f
1
2
D
1
10000
11800
2
D 0.5309
Thus from Figure 42.43(a), the series arm components are
mL
2
D
0.530919
2
D 5.04 mH
and the shunt arm comprises mC D 0.53090.0531 D 0.0282 mF in
series with
1 m
2
4m
L D
1 0.5309
2
40.5309
19 D 6.43 mH
For the half-sections a value of m D 0.6 is taken to obtain matching.
Thus from Figure 42.50,
mL
2
D
0.619
2
D 5.7mH,
mC
2
D
0.60.0531
2
0.0159 mF
5.7 mH 9.5 mH 9.5 mH
5.04 mH 5.04 mH 5.7 mH
0.0159 µF
10.13 mH
0.0531 µF
0.0282 µF
6.43 mH
0.0159 µF
10.13 mH
Half section Prototype section
"m-derived" section Half section
Figure 42.53
Filter networks 837
and
1 m
2
2m
L D
1 0.6
2
20.6
19 10.13 mH
The complete filter is shown in Figure 42.53.
Further problems on practical composite filter sections may be found in
Section 42.10 following, problems 23 and 24, page 840
42.10 Further problems
on filter networks
Low-pass filter sections
1 Determine the cut-off frequency and the nominal impedance of each
of the low-pass filter sections shown in Figure 42.54.
[(a) 1592 Hz; 5 k (b) 9545 Hz; 600 ]
2 A filter section is to have a characteristic impedance at zero frequency
of 500 and a cut-off frequency of 1 kHz. Design (a) a low-pass
T section filter, and (b) a low-pass section filter to meet these
requirements.
[(a) Each series arm 79.6 mH, shunt arm 0.637
µF
(b) Series arm 159.2 mH, each shunt arm 0.318
µF]
3 Determine the value of capacitance required in the shunt arm of a
low-pass T section if the inductance in each of the series arms is
40 mH and the cut-off frequency of the filter is 2.5 kHz.
[0.203
µF]
4 The nominal impedance of a low-pass section filter is 600 and
its cut-off frequency is at 25 kHz. Determine (a) the value of the
characteristic impedance of the section at a frequency of 20 kHz and
(b) the value of the characteristic impedance of the equivalent low-
pass T section filter. [(a) 1 k (b) 360 ]
0.5 H 0.5 H
0.04 µF
20 mH
27.8 nF 27.8 nF
(a)
(b)
Figure 42.54
5 The nominal impedance of a low-pass section filter is 600 .If
the capacitance in each of the shunt arms is 0.1
µF determine the
inductance in the series arm. Make a sketch of the ideal and the
practical attenuation/frequency characteristic expected for such a filter
section. [72 mH]
6 A low-pass T section filter has a nominal impedance of 600 and
a cut-off frequency of 10 kHz. Determine the frequency at which
the characteristic impedance of the section is (a) zero, (b) 300 ,
(c) 600 [(a) 10 kHz (b) 8.66 kHz (c) 0]
High-pass filter sections
7 Determine for each of the high-pass filter sections shown in
Figure 42.55 (i) the cut-off frequency, and (ii) the nominal
impedance.
[(a) (i) 22.51 kHz (ii) 14.14 k (b) (i) 281.3 Hz (ii) 1414 ]